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INTRODUCTION TOGEOMAGNETIC FIELDSSecond Edition

WALLACE H. CAMPBELL

PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGEThe Pitt Building, Trumpington Street, Cambridge, United Kingdom

CAMBRIDGE UNIVERSITY PRESSThe Edinburgh Building, Cambridge CB2 2RU, UK40 West 20th Street, New York, NY 10011-4211, USA477 Williamstown Road, Port Melbourne, VIC 3207, AustraliaRuiz de Alarcon 13, 28014 Madrid, SpainDock House, The Waterfront, Cape Town 8001, South Africa

http://www.cambridge.org

C© Wallace H. Campbell 1997, 2003

This book is in copyright. Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,no reproduction of any part may take place withoutthe written permission of Cambridge University Press.

First published 1997Second edition 2003

Printed in the United Kingdom at the University Press, Cambridge

Typefaces Times Roman 10/13 pt and Stone Sans System LATEX2ε []

A catalog record for this book is available from the British Library

Library of Congress Cataloging in Publication data

ISBN 0 521 82206 8 hardbackISBN 0 521 52953 0 paperback

Contents

Preface page ixAcknowledgements xii1 The Earth’s main field 1

1.1 Introduction 11.2 Magnetic Components 41.3 Simple Dipole Field 81.4 Full Representation of the Main Field 151.5 Features of the Main Field 341.6 Charting the Field 411.7 Field Values for Modeling 471.8 Earth’s Interior as a Source 511.9 Paleomagnetism 581.10 Planetary Fields 621.11 Main Field Summary 631.12 Exercises 65

2 Quiet-time field variations and dynamo currents 672.1 Introduction 672.2 Quiet Geomagnetic Day 682.3 Ionosphere 742.4 Atmospheric Motions 812.5 Evidence for Ionospheric Current 862.6 Spherical Harmonic Analysis of the Quiet Field 922.7 Lunar, Flare, Eclipse, and Special Effects 1022.8 Quiet-Field Summary 1082.9 Exercises 109

3 Solar–terrestrial activity 1113.1 Introduction 1113.2 Quiet Sun 1133.3 Active Sun, Sunspots, Fields, and Coronal Holes 1163.4 Plages, Prominences, Filaments, and Flares 1223.5 Mass Ejections and Energetic Particle Events 1273.6 Interplanetary Field and Solar Wind Shocks 1283.7 Solar Wind–Magnetosphere Interaction 1353.8 Geomagnetic Storms 139

v

vi Contents

3.9 Substorms 1423.10 Tail, Ring, and Field-Aligned Currents 1443.11 Auroras and Ionospheric Currents 1523.12 Radiation Belts 1653.13 Geomagnetic Spectra and Pulsations 1683.14 High Frequency Natural Fields 1733.15 Geomagnetic Indices 1753.16 Solar–Terrestrial Activity Summary 1843.17 Exercises 186

4 Measurement methods 1894.1 Introduction 1894.2 Bar Magnet Compass 1904.3 Classical Variometer 1944.4 Astatic Magnetometer 1954.5 Earth-Current Probe 1964.6 Induction-Loop Magnetometer 1984.7 Spinner Magnetometer 2004.8 Fluxgate (Saturable-Core) Magnetometer 2014.9 Proton-Precession Magnetometer 2034.10 Optically Pumped Magnetometer 2054.11 Zeeman-Effect Magnetometer 2104.12 Cryogenic Superconductor Magnetometer 2124.13 Gradient Magnetometer 2134.14 Comparison of Magnetometers 2154.15 Observatories 2154.16 Location and Direction 2194.17 Field Sampling and Data Collection 2194.18 Tropospheric and Ionospheric Observations 2214.19 Magnetospheric Measurements 2224.20 Instrument Summary 2254.21 Exercises 226

5 Applications 2285.1 Introduction 2285.2 Physics of the Earth’s Space Environment 2295.3 Satellite Damage and Tracking 2305.4 Induction in Long Pipelines 2335.5 Induction in Electric Power Grids 2355.6 Communication Systems 2375.7 Disruption of GPS 2395.8 Structure of the Earth’s Crust and Mantle 239

5.8.1 Surface Area Traverses for Magnetization Fields 2415.8.2 Aeromagnetic Surveys 241

Contents vii

5.8.3 Conductivity Sounding of the Earth’s Crust 2465.8.4 Conductivity of the Earth’s Upper Mantle 251

5.9 Ocean Bottom Studies 2535.10 Continental Drift 2555.11 Archeomagnetism 2575.12 Magnetic Charts 2575.13 Navigation 2585.14 Geomagnetism and Weather 2595.15 Geomagnetism and Life Forms 2625.16 Solar–Terrestrial Disturbance Predictions 2695.17 Magnetic Frauds 271

5.17.1 Body Magnets 2775.17.2 Prediction of Earthquakes 277

5.18 Summary of Applications 2785.19 Exercises 279

Appendix A Mathematical topics 280A.1 Variables and Functions 280A.2 Summations, Products, and Factorials 282A.3 Scientific Notations and Names for Numbers 282A.4 Logarithms 283A.5 Trigonometry 284A.6 Complex Numbers 285A.7 Limits, Differentials, and Integrals 287A.8 Vector Notations 289A.9 Value Distributions 291A.10 Correlation of Paired Values 294

Appendix B Geomagnetic organizations, services,and bibliography 296

B.1 International Unions and Programs 296B.2 World Data Centers for Geomagnetism 298B.3 Special USGS Geomagnetics Website 303B.4 Special Geomagnetic Data Sets 304B.5 Special Organization Services 306B.6 Solar–Terrestrial Activity Forecasting Centers 309B.7 Bibliography for Geomagnetism 312B.8 Principal Scientific Journals for Geomagnetism 313

Appendix C Utility programs for geomagnetic fields 315C.1 Geomagnetic Coordinates 1940–2005 315C.2 Fields from the IGRF Model 316C.3 Quiet-Day Field Variation, Sq 316C.4 The Geomagnetic Disturbance Index, Dst 317C.5 Location of the Sun and Moon 318

viii Contents

C.6 Day Number 318C.7 Polynomial Fitting 319C.8 Quiet-Day Spectral Analysis 319C.9 Median of Sorted Values 319C.10 Mean, Standard Deviation, and Correlation 320C.11 Demonstration of Spherical Harmonics 320C.12 Table of All Field Models 321

References 322Index 332

Chapter 1

The Earth’s main field

1.1 IntroductionThe science of geomagnetism developed slowly. The earliest writingsabout compass navigation are credited to the Chinese and dated to250 years B.C. (Figure 1.1). When Gilbert published the first textbook ongeomagnetism in 1600, he concluded that the Earth itself behaved as agreat magnet (Gilbert, 1958 reprint) (Figure 1.2). In the early nineteenthcentury, Gauss (1848) introduced improved magnetic field observationtechniques and the spherical harmonic method for geomagnetic fieldanalysis. Not until 1940 did the comprehensive textbook of Chapmanand Bartels bring us into the modern age of geomagnetism. The bibli-ography in the Appendix, Section B.7, lists some of the major textbooksabout the Earth’s geomagnetic field that are currently in use.

For many of us the first exposure to the concept of an electromagneticfield came with our early exploration of the properties of a magnet. Itsstrong attraction to other magnets and to objects made of iron indicatedimmediately that something special was happening in the space betweenthe two solid objects. We accepted words such as field, force field, andlines of force as ways to describe the strength and direction of the pushor pull that one magnetic object exerted on another magnetic materialthat came under its influence. So, to start our subject, I would like torecall a few of our experiences that give reality to the words magneticfield and dipole field.

Toying with a couple of bar magnets, we find that they will attractor oppose each other depending upon which ends are closer. This exper-imentation leads us to the realization that the two ends of a magnet have

1

2 The Earth’s main field

Figure 1.1. The Chinesereport that the compass(Si Nan) is described in theworks of Hanfucious, whichthey date between 280 and233 B.C. The spoon-shapedmagnetite indicator, balancingon its heavy rounded bottom,permits the narrow handle topoint southward, to align withthe directions carvedsymmetrically on anonmagnetic baseplate. Thisphotograph shows a recentreproduction, manufacturedand documented by theCentral Iron and SteelResearch Institute, Beijing.

Figure 1.2. Diagram fromGilbert’s 1600 textbook ongeomagnetism in which heshows that the Earth behavesas a great magnet. The fielddirections of a dip-needlecompass are indicated as tiltedbars.

1.1 Introduction 3

oppositely directed effects or polarity. It is a short but easy step for us tounderstand the operation of a compass when we are told that the Earthbehaves as a great magnet.

In school classrooms, at more sophisticated levels of science explo-ration, many of us learned that positive and negative electric chargesalso have attraction/repulsion properties. The simple arrangement oftwo charges of opposite sign constitute an electric dipole. The productof the charge size and separation distance is called the dipole moment;the pattern of the resulting electric field is called dipolelike.

The subject of this chapter is the dipolelike magnetic field that wecall the “Earth’s main field.” I will demonstrate that this magnetic fieldshape is similar in form to the field from a pair of electric chargeswith opposite signs. Knowing that a loop of current could produce adipolelike field, arguments are given to discount the existence of a large,solid iron magnet as the Earth-field source. Rather, the main field ori-gin seems to reside with the currents flowing in the outer liquid core ofthe Earth, which derive their principal alignment from the Earth’s axialspin.

To bring together the many measurements of magnetic fields thatare made on the Earth, a method has been developed for depicting thesystematic behavior of the fields on a spherical surface. This sphericalharmonic analysis (SHA) creates a mathematical representation of theentire main field anywhere on Earth using only a small table of num-bers. The SHA is also used to prove that the main field of the Earthoriginates mostly from processes interior to the surface and that only aminor proportion of the field arises from currents in the high and distantexternal environment of the Earth. We shall see that the SHA dividesthe contributions of field into dipole, quadrupole, octupole, etc., dis-tinct parts. The largest of these, the dipole component, allows us to fix ageomagnetic coordinate system (overlaying the geographic coordinates)that helps researchers easily organize and explain various geophysicalphenomena.

The slow changes of flow processes in the Earth’s deep liquid in-terior that drive the geomagnetic field require new sets of SHA tablesand revised geomagnetic maps to be produced regularly over the years.Such changes are typically quite gradual so that some of the past andfuture conditions are predictable over a short span of time. A specialscience of paleomagnetism examines the behavior of the ancient fieldbefore the Earth assumed its present form. Paleomagnetic field changes,for the most part, are not predictable and give evidence of the magnetismsource region and the Earth’s evolution.

Also, in this chapter we will look at the definitions of terms used torepresent the Earth’s main field. We will see how the descriptive maps


and field model tables are obtained and appreciate the meaning that thesenumbers provide for us.

1.2 Magnetic ComponentsA typical inexpensive compass such as a small needle dipole magnetfreely balanced, or suspended at its middle by a long thread, will alignitself with the local horizontal magnetic field in a general north–southdirection. The north-pointing end of this magnet is called the north pole;its opposite end, the south pole. Because opposite ends of magnets,or compass needles, are found to attract each other, the Earth’s dipolefield, attracting the north pole of a magnet toward the northern arcticregion, should really be called a south pole. Fortunately, to avoid suchconfusion, the convention is ignored for the Earth so that geographicand geomagnetic pole names agree. Other adjectives sometimes givenare Boreal for the northern pole and Austral for the southern pole. Wesay that our compass points northward, although, in fact, it just alignsitself in the north–south direction. The early Chinese, who first used acompass for navigation (at least by the fourteenth century) consideredsouthward to be the important pointing direction (Figure 1.1). Naturallymagnetized magnetite formed the first compasses. Early Western civi-lization called that black, heavy iron compound lodestone (sometimesspelled loadstone) meaning “leading stone.” It is believed that the word“magnet” is derived from Magnesia (north-east of Ephesus in ancientMacedonia) where lodestone was abundant.

By international agreement, a set of names and symbols is used to de-scribe the Earth’s field components in a “right-hand system.” Figure 1.3illustrates this nomenclature for a location in the Northern Hemispherewhere the total field vector points into the Earth. The term right-handsystem means that if we aligned the thumb and first two fingers of ourright hand with the three edges that converge at a box corner, then the xdirection would be indicated by our thumb, the y direction by our index(pointing) finger, and the z direction by the remaining finger. We saythese are the three orthogonal directions along the X , Y , and Z axes inspace because they are at right angles (90◦) to each other. When a mea-surement has both a size (magnitude) and a direction, it can be drawnas an arrow with a particular heading that extends a fixed distance (toindicate magnitude) from the origin of an orthogonal coordinate system.Such an arrow is called a vector (see Section A.6). Any vector may berepresented in space by the composite vectors of its three orthogonalcomponents (projections of the arrow along each axis).

A magnetic field is considered to be in a positive direction if anisolated north magnetic pole would freely move in that field direction.

1.2 Magnetic Components 5

Figure 1.3. Components ofthe geomagnetic fieldmeasurements for a sampleNorthern Hemisphere totalfield vector F inclined into theEarth. An explanation of theletters and symbols is given inthe text.

Observers prefer to describe a vector representing the Earth’s field inone of two ways: (1) three orthogonal component field directions withpositive values for geographic northward, eastward, and vertical intothe Earth (negative values for the opposite directions) or (2) the hori-zontal magnitude, the eastward (minus sign “–” for westward) angulardirection of the horizontal component from geographic northward, andthe downward (vertical) component. The first set is typically called theX , Y , and Z (XYZ-component) representation; the last set is called theH (horizontal), D (declination), and Z (into the Earth) (HDZ-component)representation (or sometimes DHZ ). In equations, a boldface on a fieldletter (e.g., H) will be used to emphasize the vector property; withoutthe boldface we will just be interested in the size (magnitude).

In the early days of sailing-ship navigation the important measure-ment for ship direction was simply D, the angle between true north andthe direction to which the compass needle points. Ancient magnetic ob-servations therefore used the HDZ system of vector representation. Bysimple geometry we obtain

X = H cos(D), Y = H sin(D). (1.1)

(See Section A.5 for trigonometric functions.) The total field strength,F (or T ), is given as

F =√

X 2 + Y 2 + Z2 =√

H2 + Z2. (1.2)


The angle that the total field makes with the horizontal plane is calledthe inclination, I, or dip angle:

Z

H= tan(I). (1.3)

The quiet-time annual mean inclination of a station, called its “DipLatitude”, becomes particularly important for the ionosphere at about60 to 1000 km altitude (Section 2.3) where the local conductivity isdependent upon the field direction.

Although the XYZ system provides the presently preferred coordi-nates for reporting the field and the annual INTERMAGNET data disksfollow this system, activity-index requirements (Section 3.13) and somenational observatories publish the field in the HDZ system. It is a simplematter to change these values using the angular relationships shown inFigure 1.3. The conversion from X and Y to H and D becomes

H =√

(X 2 + Y 2) and D = tan−1(Y/X ). (1.4)

On occasion, the declination angle D in degrees (D◦) is expressed inmagnetic eastward directed field strength D (nT) and obtained from therelationship

D (nT) = H tan (D◦). (1.5)

Sometimes the change of D (nT) about its mean is called a magneticeastward field strength, �E. (For small, incremental changes in a valueit is the custom to use the symbol �.)

In the Earth’s spherical coordinates, the three important directionsare the angle (θ ) measured from the geographic North Pole along a greatcircle of longitude, the angle (φ) eastward along a latitude line measuredfrom a reference longitude, and the radial direction, r, measured from thecenter of the Earth. On the Earth’s surface (where x, y, and z correspondto the −θ, φ, and −r directions) the field, B, in spherical coordinatesbecomes

Bθ = −X, Bφ = Y, and Br = −Z. (1.6)

Originally, the HDZ system was used at most world observatoriesbecause the measuring instruments were suspended magnets and therewas a direct application to navigation and land survey. Usually, onlyan angular reading between a compass northward direction and geo-graphic north was needed. In the HDZ system, the data from differentobservatories have different component orientations with respect to theEarth’s axis and equatorial plane. The θφr system is used for mathemat-ical treatments in spherical analysis (of which we will see more in thischapter). The X YZ coordinate system is necessary for field recordings

1.2 Magnetic Components 7

by many high-latitude observatories because of the great disparity in thegeographic angle toward magnetic north at polar region sites. The X YZsystem is becoming the preferred coordinate system for most moderndigital observatories. Computers have made it simple to interchange thedigital field representation into the three coordinate systems. Figure 1.3shows the angle of inclination (dip), I , and the total field vector, F.

The unit size of fields is a measurable quantity. We can appreciate thisfact when we consider the amount of force needed to separate magnetsof different strengths or the amount of force that must be used to push acompass needle away from its desired north–south direction. Let us notelaborate on tedious details of establishing the unit sizes of fields. Whatwill be called “field strength” results from a measurement of a quantitycalled “magnetic flux density,” B, that can be obtained from a comparisonto force measurements under precisely prescribed conditions. The unitsfor this field strength have appeared differently over the years; Table 1.1lists equivalent values of B.

Table 1.1. Equivalent magneticfield units

B = 104 Gauss

B = 1 Weber/meter2

B = 109 gamma

B = 1 Tesla

At present, in most common usage, the convenient size of mag-netic field units is the gamma, or γ , a lower-case Greek letter to honorCarl Friedrich Gauss, the nineteenth-century scientist from Gottingen,Germany, who contributed greatly to our knowledge of geomagnetism.The International System (SI) of units, specified by an agreement ofworld scientists, recommends use of the Tesla (the name of an earlypioneer in radiowave research). With the prefix nano meaning 10−9, ofcourse, one gamma is equivalent to one nanotesla (nT), so there shouldbe no confusion when we see either of these expressions. To familiar-ize the reader with this interchange (which is common in the presentliterature), I will use either name at different times in this book.

Geomagnetic phenomena have a broad range of scales. The mainfield is nearly 60, 000 (6 × 104; see scientific notation in Section A.3)gamma near the poles and about 30, 000 (3 × 104) gamma near theequator. A small, 2 cm, calibration magnet I have in my office is1 × 108 gamma at its pole (about 10,000 times the Earth’s surface fieldin strength). Quiet-time daily field variations can be about 20 gammaat midlatitudes and 100 gamma at equatorial regions. Solar–terrestrial


disturbance–time variations occasionally reach 1,000 gamma at the au-roral regions and 250 gamma at midlatitudes. Geomagnetic pulsationsarising in the Earth’s space environment are measured in the 0.01 gammato 10 gamma range at surface midlatitude locations. In Chapter 4 we willsee how this great 106 dynamic range of the source fields is accommo-dated by the measuring instruments.

The magnetic fields that interest us arise from currents. Currentscome from charges that are moving. Much of the research in geomag-netism concerns the discovery (or the use of) the source currents re-sponsible for the fields found in the Earth’s environment. Then, we ask,what about the fields from magnetic materials; where is the current tobe found? A simple “Bohr model” (with planetarylike electrons about asunlike nucleus) suffices in our requirements for visualizing the atomicstructure. In this model the spinning charges of orbital electrons in theatomic structure provide the major magnetic properties. Most atoms innature contain even numbers of orbiting electrons, half circulating inone direction, half in another, with both their orbital and spin magneticeffects canceling. When canceling does not occur, typically when thereare unpaired electrons, there is a tendency for the spins of adjacentatoms or molecules to align, establishing a domain of unique field di-rection. Large groupings of similar domains give a magnet its specialproperties. We will discuss this subject further in Section 4.2 on geo-magnetic instruments. However, for now, we find consistency in the ideathat charges-in-motion create our observed magnetic fields.

1.3 Simple Dipole FieldTo many of us, the first exposure to the term “dipole” occurred in learn-ing about the electric field of two point charges of opposite sign placeda short distance from each other. Figure 1.4 represents such an arrange-ment of charges, +q and −q (whose sizes are measured in units called“coulombs”), separated by a distance, d, along the z axis of an orthog-onal coordinate system. We call the value (qd ) by the distinctive namedipole moment and assign it the letter “p.” The units of p are coulomb-meters. Figure 1.5 shows the dipole as well as symmetric quadrupoleand octupole arrangements of charge at the corners of the respectivefigures. The reason for introducing the electric charges and multipoleshere is to help us understand the nomenclature of the magnetic fields,for which isolated poles do not exist, although the magnetic field shapesare identical to the shapes of multipole electric fields.

The point P(x, y, z) is the location, for position x, y, and z from thedipole axis origin, at which the electric field strength from the dipolecharges is to be determined (Figure 1.4). This location is a distance

1.3 Simple Dipole Field 9

Figure 1.4. Electric dipole ofcharges, ±q, and thecorresponding coordinatesystem. An explanation of theletters and symbols is given inthe text.

Figure 1.5. Chargedistribution for the electricdipole, quadrupole, andoctupole configurations.

r =√

x2 + y2 + z2 from the midpoint between the two charges and at anangle θ from the positive Z axis. We will call this angle the colatitude(colatitude = 90◦− latitude) of a location. The angle to the projectionof r onto the X –Y plane, measured clockwise, is called φ. Later wewill identify this angle with east longitude. Sometimes we will see theletter e, with the r, θ , or φ subscript, used to indicate the respective unitdirections in spherical coordinates. Obviously, there is symmetry aboutthe Z axis so the electric field at P doesn’t change with changes in φ.Thus it will be sufficient to describe the components of the dipole fieldsimply along r and θ directions.

Now I will need to use some mathematics. It is necessary to showthe exact description that defines something we can easily visualize: theshape of an electric field resulting from two electric charges of oppositesign and separated by a small distance. I will then demonstrate that sucha mathematical representation is identical to the form of a field from acurrent flowing in a circular loop. That proof is important for all ourdescriptions of the Earth’s main field and its properties because we willneed to discuss the source of the main dipole field, global coordinatesystems, and main-field models. If the mathematics at this point is toodifficult, just read it lightly to obtain the direction of the developmentand come back to the details when you are more prepared.


Let us start with a property called the electric potential, , of a pointcharge in air from which we will subsequently obtain the electric field.

= q

4π ∈0 r, (1.7)

where q is the coulomb charge, r is the distance in meters to the observa-tion, ∈0 (the “inductive capacity of free space”) is a constant typical ofthe medium in which the field is measured, and is measured in volts.

For two charges with opposite signs, separated by a distance, d, thepotential at point r at x, y, z coordinate distances becomes

= 1

4π ∈0

[q√

(z − d/2)2 + x2 + y2+ −q√

(z + d/2)2 + x2 + y2

](1.8)

Now, for the typical dipole, d is very small with respect to r so, withsome algebraic manipulation, we can write

= q

4π ∈0 r

[(1 + zd

2r2

)−

(1 − zd

2r2

)]+ �A, (1.9)

where �A represents terms that become negligible when d = r. Because(z/r) = cos(θ ), Equation (1.9) can be written in the form

= qd cos(θ )

4π ∈0 r2. (1.10)

Now let us see the form of the electric field using Equation (1.7).We are going to be interested in a quantity called the gradient or grad(represented by an upside-down Greek capital delta; see Section A.8) ofthe potential. In spherical coordinates the gradient can be represented bythe derivatives (slopes) in the separate coordinate directions:

∇ = er

(δ

δr

)+ eθ

(δ

rδθ

)+ eφ

(1

r (sin θ )

δ

δφ

), (1.11)

where the es are the unit vectors in the three spherical coordinate direc-tions, r, θ , and φ. The electric field, obtained from the negative of thatgradient, is given as

Er = − δ

δr= p

2π ∈0

(cos θ

r3

)er (1.12)

and

Eθ = −1

r

δ

δθ= p

4π ∈0

(sin θ

r3

)eθ , (1.13)

where p is the electric dipole moment qd.Symmetry about the dipole axis means doesn’t change with angle

φ, so E in the φ direction is zero. Equations (1.12) and (1.13) define theform of an electric dipole field strength in space. If we would like to drawlines representing the shape of this dipole field (to show the directions


Figure 1.6. Electric dipole (ofmoment p) field configurationwith directions for thecomponents of electric fieldvectors E in the r and θ

directions to an arbitraryobservation point P ; h is theequatorial field line distance.

that a charge would move in its environment), there is a convenientequation,

r = h sin2(θ ), (1.14)

that can be used, in which h is the distance from the dipole center to theequatorial crossing (at θ = 90◦) of the field line (Figure 1.6).

These field descriptions (Equations (1.11) to (1.13)) come from sci-entists, mainly of the late eighteenth and early nineteenth centuries, whointerpreted laboratory measurements of charges, currents, and fieldsto establish mathematical descriptions of the natural electromagnetic“laws” they observed. At first there was a multitude of laws and equa-tions, covering many situations of currents and charges and relatingelectricity to magnetism. Then by 1873, James Clerk Maxwell broughtorder to the subject by demonstrating that all the “laws” could be derivedfrom a few simple equations (that is, “simple” in mathematical form). Forexample, in a region where there is no electric charge, Maxwell’s equa-tions show that there is no “divergence” of electric field, a statement thatmathematical shorthand shows as

∇ · E = 0, (1.15)

where the “del-dot” symbol is explained in Section A.8. But E is givenas – grad (which is titled “the negative gradient of the scalar potential”).Thus, for the mathematically inclined, it follows that

∇ · E = −∇∇ = 0 (1.16)

or

∇2 = 0, (1.17)


for which Equation (1.10) can be shown (by those skilled in mathematicalmanipulations) to be a solution for a dipole configuration of charges.

A simple experiment, often duplicated in science classrooms, is toconnect a battery and an electrical resistor to the ends of an iron wire(with an insulated coating) that has been wrapped in a number of turns,spiraling about a wooden matchstick for shape. It is then demonstratedthat when current flows, the wire helix behaves as if it were a dipolemagnet aligned with the matchstick, picking up paper clips or deflectinga compass needle. If the current direction is reversed by interchangingthe battery connections, then the magnetic field direction reverses.

Now let us illustrate with mathematics how a current flowing in asimple wire loop produces a magnetic field in the same form as the elec-tric dipole. My purpose is to help us visualize a magnetic dipole, whenthere isn’t a magnetic substance corresponding to the electric charges,so that we can later understand the origin of the main field in the liquidflows of the Earth’s deep-core region.

Consider Figure 1.7, in which a current, i, is flowing in the X –Y planealong a loop enclosing area, A, of radius b, for which Z is the normal(perpendicular) direction. Let P be any point at a distance, r, from theloop center and at a distance, R, from a current element moving a distance,ds. The electromagnetic law for computing the element of field, dB, fromthe current along the wire element, ds, is

dB = µ0i

4π

ds R sin(α)

R3, (1.18)

where α is the angle between ds and R, so that dB is in the direction that

Figure 1.7. Coordinatesystem for a loop of current i ,having radius b, area A, andenclosed perimeter of elementlength ds. The magnetic fieldvectors of B in the r and θ

directions with respect to anorthogonal (right-hand) x, y, zcoordinate system are shown.


a right-handed screw would move when turning from ds (in the currentdirection) toward R (directed toward P). The magnetic properties of themedium are indicated by the constant, µ0, called the “permeability offree space.” We wish to find the r and θ magnetic field components of B,at any point in space about the loop, with the simplifying conditions thatr � b. Using the electromagnetic laws, we sum the dB contributions tothe field for each element of distance around the loop, and after somemathematics obtain

Br = µ0i A cos(θ )

2πr3(1.19)

and

Bθ = µ0i A sin(θ )

4πr3. (1.20)

Comparing these two field representations with those we obtained forthe electric dipole (Equations (1.12) and (1.13)), we see that the samefield forms will be produced if we let the current times the area (i A)correspond to p, the electric dipole moment, qd. Thus, calling M themagnetic dipole moment,

M = iA (1.21)

or

M = md, (1.22)

where d becomes the equivalent separation of hypothetical magneticpoles of strength m.

We saw, in the parallel case of the electric dipole, that E was obtainedfrom the scalar potential in a charge-free region. In a similar fashion,Maxwell’s equations show that

B = −∇V, (1.23)

where V is called the magnetic scalar potential. Then, a person with mathcompetence can write, for a current-free region (where curl B = 0),

∇2V = 0 (1.24)

and obtain a dipole solution

V = µ0M cos(θ )

4πr2. (1.25)

To a first approximation, the Earth’s field in space behaves as amagnetic dipole. At the Earth’s surface we call r = a. Then

Br = −Z = 2[µ0M cos(θ )]

4πa3= Z0 cos(θ ) (1.26)


and

Bθ = −H = µ0M sin(θ )

4πa3= H0 sin(θ ), (1.27)

where constant H0 = Z0/2. The total field magnitude, F, is just

F =√

H2 + Z2. (1.28)

On average, about ninety percent of the Earth’s field is dipolar so we canuse the approximation, H0 = 3.1 × 104 gamma, for rough field model-ing. In Equations (1.26) and (1.27), recall (Section A.5) that sin(90◦) =1, sin(0◦) = 0, cos(90◦) = 0, and cos(0◦) = 1. For the SouthernHemisphere, where 90◦ < θ ≤ 180◦, note that sin(180◦ − θ ) = sin(θ )and cos(180◦ − θ ) = − cos(θ ). So the magnitude of the Earth’s fieldat the equator (θ = 90◦) is just H0 and at the poles (θ = 0◦ or 180◦)just 2H0.

For the dipole, the inclination, I (direction of the Earth’s field awayfrom the horizontal plane), at any θ is defined from

tan(I) = Z

H= 2 cot(θ ). (1.29)

This is a valuable relationship for measurements of continental drift(see Section 5.10). It means that we can determine our geomagneticlatitude (90◦ − θ ) from field measurements of H and Z. Using ancientrocks to tell the field direction in an earlier geological time, the ap-parent latitude of the region can be fixed by Equation (1.29). Later, inSection 1.9, there will be more details regarding this paleomagnetismsubject.

Conjugate points on the Earth’s surface are locations P and P′ thatcan be connected by a single dipole field line (Figure 1.8). The relativelystrong Earth’s field lines become guiding tracks for charged particles inthe magnetosphere. The positions for conjugate points are used in studiesof the Earth arrival of these phenomena from distant locations in space.The dipole field lines will extend out into the equatorial plane a distance,re. Up to about 65◦ geomagnetic latitude, θ ′ (in degrees), the length ofthis field line can be approximated by the relationship

length ≈ 0.38θ ′re (1.30)

where the length and re are in similar units (e.g., kilometers or Earthradii). Figure 1.9 shows the relationship of latitude and field-line equa-torial distance. As an illustration, at 50◦ geomagnetic latitude, read theappropriate x-axis scale; move vertically to the curve intersection, thenread horizontally to the corresponding y-axis scale, obtaining 2.5 Earthradii for the distant extent of that field line. We shall see, in Chapter 3,that the outermost field lines of the Earth’s dipole field are distorted

1.4 Full Representation of the Main Field 15

Figure 1.8. Geomagneticlocations, based on a sphericalcoordinate system alignedwith respect to the dipolefield, with latitude θ ′ = 90 − θ

(where θ is the colatitude) andlongitude φ. A dipole field lineof length l , connecting theconjugate points at P and P ′,extends to a distance re in theequatorial plane from thedipole center 0.

by a wind of particles and fields that arrive from the Sun; such changebecomes quite noticeable above 60◦.

The magnetic shell parameter, L shell, is an effective mean equa-torial radius of a magnetic field shell, which, for a given field strength,B, defines the trapped-particle flux in the space about the Earth. Com-putation of the L shells for the Earth’s field is complex. However, fora dipole field, the L-shell values may be considered almost equiva-lent to the number of Earth radii that the field line extends into space,re, and is a good approximation for all but the high latitudes. Theinvariant latitude (in degrees) of a location is obtained from L by therelationship

cos (invariant latitude) = 1√L

(1.31)

Figure 1.10 shows polar views of the L-shell contours for the twohemispheres, computed for the model, extremely quiet field of 1965.Many of the high-latitude geomagnetic phenomena are best orga-nized when plotted with respect to their L shell or invariant latitudepositions.

1.4 Full Representation of the Main FieldNow comes the most difficult part of this book, the representation ofthe main field by equations and tables. There is a considerable amount


Figure 1.9. Equatorial radialextent re (from the Earth’scenter) of a dipole field linestarting from latitude θ ′ at theEarth’s surface.

of mathematics here, with analysis techniques and shorthand mathsymbols that can frighten the casual reader. I will try to step gentlyin this section, but it is necessary for us to go through the details be-cause there will be so many important physical results we can prop-erly appreciate later if we understand their origin in the main fieldrepresentation.

We will start with some of Maxwell’s equations and show how therelationships appear in a spherical coordinate system. Then we will lookfor a solution of the equations of a type that will let us separate currentsources that arise above and below a sphere’s surface. Next, we will lookat a method for fitting the measurements from a surface of observatoryfield values into the equations that produce our Earth’s field models. Itis important to know some of the strengths and failings of the methodsso that we understand their successful application in geophysics. Wewill also find the main field representation important in the chapter on

Figure 1.10. L -shell contours,computed for 100-kmaltitude, in the Northern (top)and Southern (bottom)Hemisphere regions.Geographic east and westradial longitude lines andcircles of latitude (from 30◦ tothe pole) are shown. TheseL -values were computed forthe extremely quiet year,1965, when there was aminimum distortion of thepolar contours by solar wind.


quiet-field variations as well as in our discussion of solar–terrestrialdisturbances. If you are not ready for the mathematics at this time, atleast read through the steps lightly to focus upon what is being computedand the sequence followed.

Maxwell’s great contribution to the understanding of electromag-netic phenomena was to show that all the measurements and lawsof field behavior could be derived from a few compact mathematicalexpressions. We will start with one of these equations, adjusted forthe assumptions that only negligible electric field changes occur andthat the amount of current flowing across the boundary between theEarth and its atmosphere is relatively insignificant. Then, at the Earth’ssurface

∇ × B = i

(δBz

δy− δBy

δz

)+ j

(δBx

δz− δBz

δx

)+ k

(δBy

δx− δBx

δy

)= 0, (1.32)

where i, j, k represent the three orthogonal directions and δ indicates that“partial” derivatives are used (see Section A.7). This equation is read “thecurl of B equals zero” and requires that the field can be obtained fromthe “negative gradient of a scalar potential” so

B = −[

iδV

δx+ j

δV

δy+ k

δV

δz

]= −∇V. (1.33)

The other Maxwell’s equation that we will use is

∇ · B =[

δBx

δx+ δBy

δy+ δBz

δz

]= 0. (1.34)

This equation is read “the divergence of the field is zero.” Now, puttingEquations (1.33) and (1.34) together, we obtain

∇ · ∇V = ∇2V = 0, (1.35)

which is read as “the Laplacian of scalar V is zero.” This potentialfunction will be valid over a spherical surface through which cur-rent does not flow. In spherical coordinate notation, Equation (1.35)becomes

δ

δr

(r2 δV

δr

)+ 1

sin θ

δ

δθ

(sin θ

δV

δθ

)+ 1

sin2 θ

δ2V

δφ2= 0, (1.36)

in which r, θ , and φ are the geographic, Earth-centered coordinates ofthe radial distance, colatitude, and longitude, respectively.

Now, the solution (i.e., solving the equation for an expression of Vby itself ) that is sought is one that is a product of three expressions.The first of these expressions is to be only a function of r; the second,

1.4 Full Representation of the Main Field 19

only a function of θ ; and the third, only a function of φ. That is whatmathematicians call a “separable” solution of the form

V (r, θ, φ) = R(r) · S(θ, φ), where S(θ, φ) = T(θ ) · L(φ). (1.37)

A solution of the potential function V for the Earth’s main field satisfyingthese requirements has the converging series of terms (devised by Gaussin 1838)

V = a∞∑

n=1

[( r

a

)nSe

n +(a

r

)n+1Si

n

], (1.38)

where the∑

means the sum of terms as n goes from 1 to an extremelylarge number, and for our studies, a is the Earth radius, Re. The seriessolution means that for each value of n the electromagnetic laws areobeyed as if that term were the only contribution to the field. We willsoon see that solving this equation for V allows us to immediately recoverthe strength of the magnetic field components at any location about theEarth.

There are two series for V . The first is made up of terms in rn. Asr increases, these terms become larger and larger; that means we mustbe approaching the current source of an external field in the increasingr direction. These terms are called Ve, “the external source terms of thepotential function” (and our reason for labeling the Sn functions with asuperscript e). By a corresponding argument for the second series, the(1/r)n terms become larger and larger as r becomes smaller and smaller,which means we must be approaching the current source of an internalfield in the decreasing r direction. Scientists call these terms Vi, “theinternal source terms of the potential function” (the reason for labelingthe corresponding Sn functions with a superscript i).

The S(θ, φ) terms of Equation (1.37) represent sets of a special classof functions called Legendre polynomials (see Section C.11) of the inde-pendent variable θ that are multiplied by sine and cosine function termsof independent variable φ. I shall leave to more detailed textbooks theexplanation of what is called the required “orthogonality” properties and“normalization” and simply define the “Schmidt quasi-normalized, as-sociated Legendre polynomial functions” that are used for global fieldanalysis. Here I will abbreviate these as Legendre polynomials, Pm

n (θ ),realizing they are a special subgroup of functions. The integers, n and m,are called degree and order, respectively; n has a value of 1 or greater,and m is always less than or equal to n.

When V is determined from measurements of the field about theEarth, analyses show that essentially all the contribution comes from theVi part of the potential function expansion. For now, let us just call this

Fig

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